Can you be hypnotized with your eyes open?
Yes! While it's common for people to close their eyes while being hypnotized, it's not a requirement.
You have probably seen examples of people being in hypnosis with their eyes open, just think of every stage hypnosis show you have ever seen. As active as some of the shows are, it's critical that people be able to see where they are going as they run around the stage being "the world's greatest race car driver" or something similar.
During the "induction", the process of transitioning into hypnosis, the eyes are commonly closed. However that's just to help the subject concentrate and is completely optional. Once the person is in hypnosis, eyes open or closed doesn't matter.
In a therapy environment, most people close their eyes during the induction and keep them closed unless they are specifically instructed to open them for some reason by the hypntherapist. However, in my experience, about 20% of subjects will spontaneously open their eyes partially or fully during the session. If you are a new hypnotherapist, it's a little disconcerting when it happens as the subject has a bit of a "lights are on but no one is home" look to them, but after a while you get used it.
In general, hypnosis has no dependence on the state of the eyes. Open, closed, whatever works for you!